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No, it shouldn't. Heading is an integral part of the game. Moreover, eliminating heading from the game won't solve the problem of traumatic brain injuries in the game. (Properly) heading the ball is way down on the list of what causes head injuries and brain trauma in football/soccer. Concussions arise largely from head to head collisions, head to other body part collisions, head to ground collisions, head to goal post collisions, and inadvertent head to ball collisions. Properly heading the ball does not transmit much force to the brain. Those studies (and there have been a number of them) tend to be a bit dubious. There might be something there, then again, there might not. As mentioned above, heading is far from the leading cause of traumatic brain injury in the game. It is very hard to control for the damage caused by those more serious head injuries. AFAIK, about the only thing that is conclusive is that players who have suffered a serious head injury should avoid heading the ball because now even the small and nominally reparable damage that results from heading can add insult to the existing injury. Helmets do exist, and some professionals who have suffered serious head injury do wear them now to prevent accumulative injuries from heading. As far as kids go, there are a number of reasons not to teach young children to head the ball. Their skulls are not fully developed, they don't have the requisite mind/body skills needed to properly head the ball, they don't have the situational awareness to identify conditions where heading isn't safe, and they tend to play bunch ball that makes for unsafe heading conditions. The skill needs to be built up gradually so that heading does become an integral part of the game by the time they are 18 or so.

In this case, they are. Or at least a significant fraction of them are. The far right's global warming skepticism is worse than mere ignorance. It is willful ignorance. Everyone is subject to irrational behaviors. The right thing to do is not to proudly wear those irrational beliefs and behaviors as a badge on ones chest. Being intentionally stupid is not something to be proud about. It is something to be ashamed of. The right thing to do to be ever vigilant against those irrational beliefs and to stop them in their tracks when one sees that one is exhibiting such beliefs and behaviors.

Maybe not you specifically, but you western Europeans have your share of irrationalisms. The modern day adherents of the body of anti-knowledge started by Ned Ludd (one of yours, not ours) include the anti-vaccination crowd, the anti-nuclear crowd, the anti-GMO crowd -- those are largely European ideas with predominantly European followers. Yes, they're here too, but in smaller numbers than in Europe. Everyone is subject to irrational behaviors. It's in our genes. Our predecessors that jumped at waving grass survived on the few occasions where that waving grass was caused by a predator rather than by the wind. Our non-predecessors who learned to ignore waving grass because it always appeared to be caused by the wind had the unfortunate outcome of being eaten by predators. As rationalizing beings, we have taken these built-in responses of seeing man-eating monsters when it truly is just the wind many steps further. We fabricate fears where there is nothing to fear, we don't fear things that we should. We as a species are in general quite terrible at estimating and dealing with risk.

First off, I am completely and utterly unimpressed by cleverbot. Is it a bot? Yes. Obviously. Is it clever? No. It cannot hold a meaningful conversation. It's very good at giving smug, smarmy, off-topic answers, but that's only because that is what is has been programmed to do. As for your "iProgrammer", the answer depends on what you mean by "give watson a full rundown of the system a computer language will be minipulating, and tells watson what he/she wants a program to do." In one sense, such systems already exist; in fact they've existed for half a century. A computer executes machine instructions. (Arguably, they don't even execute machine instructions any more. They execute microcode. A separate microprogram exists for each machine instruction.) Programmers in general haven't written in machine language for a long, long time. Most of us don't even write assembly. We instead write a full rundown of the system that tells the computer what we want it to do in terms of some third or fourth generation language. In this view, your watson already exists. It's the compiler or interpreter that translates our high level specification into something the computer can execute. In another sense, it ain't gonna happen. What you are describing is the now discredited waterfall model. It didn't work when humans performed that magical step of translating requirements to code. Adding some magical oracle (your "iProgrammer") won't help because the problem with the waterfall model was that we don't know how to give that "full rundown of the system".

Simple. People everywhere for the most part are not rational beings. We are instead rationalizing beings. A corollary is that irrationality, anti-intellectualism, and technophobia are rampant and know no political bounds. The Kyoto protocol exempts nations such as India and China from compliance. Of course peoples in those countries ascribe to global warming. It jibes perfectly with their own self interests. Those countries, along with other developing nations, stand to benefit greatly from increased controls on the developed worlds. What about western Europe? The solution to global warming is increased governmental controls over people and industry. This jibes perfectly with the authoritarian left political beliefs that dominate throughout western Europe. Even though global warming is science, it comports well with those aspects western European thought that are markedly anti-science / anti-technology. Western Europeans did get it right with regard to global warming, but that isn't because they're more enlightened than anyone else. It's because their irrational beliefs forced them along a path where they would inevitably to get it right. Finally, what about the US? Global warming skeptics are almost exclusively from the radical right. This group leans more toward libertarianism as opposed than authoritarianism (except where libertarianism conflicts with other beliefs). Global warming strongly conflicts with the libertarian (minimal government), Ayn Randian ("smokestacks are beautiful"), paleo-orthodoxical religious views ("God made man to have dominion over the Earth") that predominate this group. This group is particularly susceptible to anti-intellectualism because, in their minds, they know that science oftentimes is dead wrong. Evolution is false, cosmology is false, astronomy is false. In their minds, global warming is yet another example of science getting it wrong.

So, getting back on topic, The above post is from that silly thread that was split off from this one. I quoted it here in this thread because the quoted post, rather than all of the above nonsense, is the Andromeda Paradox. So, what does the Andromeda Paradox add to the age-old and never ending philosophical debate on determinism, free will, and all that? Nothing. At least that is my opinion. Per the physics as we know it, the only events I can see / observe / measure are those that are on or inside my past light cone. The only events I can somehow influence are those that are on or inside my future light cone. My "now" lies in that no-mans land of uninfluenceable, unobservable events that lies between my past and future light cones. There is no way I, a mere mortal who is subject to the laws of physics, can know of the existence of that Andromedan invasion fleet. I could know of that fleet if instead I was an omniscient, omnipresent, supernatural being. Posit the Andromedan invasion fleet mandates supernatural knowledge, particularly omniscience and omnipresence. The Andromeda Paradox is just the age-old debate on determinism, free will, and all that writ relativistically.

Nothing. What the Andromeda paradox does say is that the point in space-time that I label as (r=2.5e6 light years, t=0) might be labeled differently by you, even if you and I are right next to one another. What it says with regard to free will, determinism, and all that philosophical claptrap-- not one thing. Adding the Andromedan invasion fleet to the paradox is not science. The vote by the Andromedan council, the launch of the invasion fleet, and its subsequent recall -- that knowledge requires omniscience and omnipresence. Posit omniscience and omnipresence and of course we're back to square one in the age old debate of free will versus determinism. Restrict knowledge to that accessible to science and the council, the launch, and the recall all vanish. That knowledge is unknowable to science.

Wrong, wrong, and wrong, xyzt. The velocity of the Andromeda galaxy with respect to us has absolutely nothing to do with the Andromeda paradox. Why are you so intransigent on this? Google (or Bing, if that's your preference) is your friend. Use it. Look for other resources than that one somewhat lousy one you insist on using (and misinterpreting). Just to reiterate, the velocity of the Andromeda galaxy with respect to us has absolutely nothing to do with the Andromeda paradox. Not one single thing. The Andromeda paradox is about how two co-located observers have different surfaces of simultaneity, and how those different surfaces inherently result in significantly different interpretations of "now" at very remote locations. This post, along with the four preceding posts, do not belong in this thread. They belong in that other thread that expands on xyzt's complete misunderstanding of the Andromeda paradox.

Emphasis mine: Replace your use of "measure" with "calculate" and you'll have a good description of the paradox. I can calculate my now, but I cannot measure it. And I certainly cannot know what happens in that now. Alternatively, replace your use of "observer" with "omniscient, omnipresent being" and you'll also have a good description of the paradox. The only thing that could measure my "now" at a distance of 2.5 million light years (or any distance, for that matter) is an omniscient, supernatural being.

This thread needs to be split into two different threads, one containing the original post and perhaps one or two other posts that address the Andromeda paradox. All the rest need to be put into a new thread, "Misinterpretation of the Andromeda paradox". xyzt, you have received negative rep (none from me) because you have derailed this thread with your misinterpretation of the paradox and because of your intransigence.

No. You have hijacked this thread with your complete misunderstanding of the Andromeda paradox. It has nothing, absolutely nothing, to do with the relative velocity between the Andromeda galaxy and the Earth.

This is not the Andromeda paradox. No, you don't. The Andromeda paradox is about two observers who are co-located but are moving with respect to one another. It is a question about the relativity of simultaneity. Iggy posted the equation wrong but has the results correct. It's clearly a typo. Here's the calculation: http://www.wolframalpha.com/input/?i=%28%282.3e22+m%29*%281.3+m%2Fs%29%2F%28speed+of+light%29%5E2%29%2Fsqrt%281-%28%281.3+m%2Fs%29%2F%28speed+of+light%29%29%5E2%29

xyzt, Iggy is right. You should retract your downvote on Iggy's post. The Andromeda paradox has absolutely nothing to do with the relative velocity between the Earth and the Andromeda galaxy. It is about how the planes of simultaneity for two different observers can differ significant at large distances even if the relative velocity between the two observers is small. Going back to the original post, The two observers' planes of simultaneity are tilted with respect to one another. The angle is small for non-relativistic velocities. Given that we're talking walking speed, the angle is very very small. However, at a far enough distance, this small angle will still result in large separations. What this means is that it doesn't make sense to talk about a universal now. There is no such thing. Your concept of "now", your plane of simultaneity, is not necessarily the same as mine. It has absolute no meaning with regard to free will, metaphysics, or any of that other philosophical excrement (see the ongoing thread "Is Philosophy crap?")

It really does, ydoaPs. Philosophically speaking, you should try looking at your posts from an objective light. It's not a pretty picture.

Exactly. (Well, almost exactly. See below on my own interpretation on the interpretations of quantum mechanics.) Metaphysics is not science. The best thing to do is to eliminate its use in the sciences. Divorcing science from philosophy was a key aspect of the scientific revolution. ydoaPs touched on that back in post #160: What is this shift you are talking about? The Aristotelian / Ptolemaic point of view remained the dominant explanation of the universe up until Newton's time. Copernicus represents the starting point rather than the end point of the divorce between science and philosophy. Newtonian mechanics represents the filing of the divorce papers. It took another few hundred years to finalize the divorce. Copernicus removed but one metaphysical assumption from that dominant Aristotelian / Ptolemaic point of view, that the Earth was the center of the universe. Copernicus kept the metaphysical baggage of circular motion as being the most pure kind of motion. Kepler took the next step, but his empirical laws were still laden with that music of the spheres metagarbage. Aristotelian physics was finally eliminated from the picture when Newton showed that Kepler's laws follow from mathematics. Even Newton's physics had its bits of metaphysical baggage. (Newton was a religious whacko even by the standards of his own time.) It took another few hundred years to eliminate that metaphysical baggage from classical mechanics. And then came quantum mechanics. To me, the right interpretation of QM is "shut up and calculate (rev 2.0)": Stop arguing about which interpretation of QM is correct. Every valid interpretation of QM yields the same predictions for the outcome of an experiment. All interpretations of QM are fundamentally flawed because every single one of them is laden with some sort of metaphysical baggage. The correct, metaphysics-free interpretation of QM is yet to be invented. Until then, just shut up and calculate, using whatever interpretation you like.

A big part of the problem is that you are being sloppy. One problem right off the bat: Are you supposed to expand that expression or are you supposed solve or solve for α? I assume it's the latter. How can you expand this? It's already expanded! Another sign of sloppiness is a consistent lack of consistency. The title says the coefficients of sin(α) and cos(α) are -a and b, respectively. Later you use a1 and a2, later yet c1 and c2. Yet another sign of sloppiness is a bunch of undefined terms. What are beta, gamma, delta, and epsilon supposed to represent? How are they related to your original expression? And what's with that partial? Where did that come from? Be consistent, do a better job of telling us (and hence telling yourself) what it is you are trying to do, and define your terms. A lot of your problems will vanish when you do these things.

That's true only if the initial conditions, the relevant physical constants, and the underlying differential equations are all perfectly known. Initial conditions and physical constants are limited by our ability to measure things. Regarding the differential equations, Newton's law of gravitation is but an approximation, and the same is most likely true for general relativity. Integrate too far into the future and that arbitrary accuracy prediction will be 100% pure fiction.

You are creating a false dichotomy here. Your diagrams, along with much that you have said in this thread, implies that "math boffs" are incapable of being creative thinkers. That in turn is motivated by your view that mathematics is a rigid world of numbers, canned formulae, and not much else. Neither one is the case. The fact is that physics and mathematics have long been intertwined. Sometimes physics has pushed the boundaries of mathematics. Other times new mathematics have allowed physicists to push the boundaries of physics. Your rigid view of mathematics as comprising numbers, canned formulae, and not much else is simply not the case. In fact, professional mathematicians can go for months, years, and entire career without using algebraic formulae. The only times professional mathematicians do touch numbers and formulae are when they have to do something mundane such as balancing their bank account statement or paying a restaurant bill. There is plenty of room in the sciences for people who think calculus is hard mathematics. (It isn't. Calculus is what budding mathematicians and physicists learn when they first enter college, if not earlier.) Vast portions of the life sciences and social sciences are largely void of mathematics. Not in physics, however. There is no escaping the deep entanglement between physics and mathematics. If you aren't doing math you aren't doing physics.

What are you talking about? "All" books? Name one. Nobody that knows orbital dynamics claimed that binary stars cannot have planets, which is apparently what you are claiming. Perhaps you are confusing "chaotic" with "unstable". They are not the same.

There are a number of hidden assumptions here that many of our outsider speculators make that are completely false. One is that the current system is moribund, decrepit, and hostile to change. That is not true. It is very lively, very active, and embraces new ideas. Another is that someone educated by this moribund, decrepit is bound to think inside the box. That too is not true. Young scientists must think out of the box. A PhD thesis must in some way extend the boundaries of current knowledge. The only way one can think out of the box is if one knows what the box is.

Is this one question or two? It looks like two separate questions to me, one about the time complexity of that loop construct, the other about the knapsack problem. If that's the case, the first question is easy. How many times is the body (not shown) of the innermost loop going to be executed? Regarding the second question, if it is a second question: You need to give us a lot more to go on. Note very well: We do notdo your homework for you at this site. Instead we help you do your own homework. This means you need to show some work, and also show some respect by not writing so vaguely.

They can. Darwin's finches, for example. Related: Google the term "ring species".

The N-body problem is chaotic, which means that even a quantum level of uncertainty will eventually lead to instability. The original post said "exact" and I took that at its word. Sans perfect information, there is no "exact" solution to the N-body problem.

Hmm. Do you have me on ignore? I answered that question in post #4.

No. You aren't doing fixed point iteration here, and you aren't touching y2.